RefertoExercise3.45andthe mgf m(t) = et+2t2~2 for anormaldistribution.For n indepen- dentobservations {Yi} from anarbitrarydistributionwithmean and variance 2,

RefertoExercise3.45andthe mgf m(t) = eμt+σ2t2~2 for anormaldistribution.For n indepen-

dentobservations {Yi} from anarbitrarydistributionwithmean μ and variance σ2, let m(t)

bethe mgf of thestandardizedrandomvariable Zi = (Yi − μ)~σ.

(a) Explainwhythe mgf of º

n(Y − μ)~σ is [m(t~

º

n)]n.

(b) Showthat

(c) Sinceas n → ∞, lim an = b implies that lim(1 + an~n)n = eb, explainwhythe mgf of º
n(Y − μ)~σ convergesto et2~2, whichisthe mgf of aN(0,1)distribution.Thesequence of mgf ’s convergingimpliesthatthesequenceof cdf ’s convergestothe N(0, 1) cdf bya continuitytheorem, sothesestepsprovetheCentralLimitTheorem.

Transcribed Image Text:

1/2 tE(Z3) n 2 3! n [m(t/n)]" = "= [1+1 (1+